尺寸为$p^2$的环和虚二次域上的格上的码
Codes over rings of size $p^2$ and lattices over imaginary quadratic fields
摘要：
设$\ell>0$是3模4的无平方整环，$Ø_K$是虚二次域$K=Q(\sqrt{-\ell})$的整数环.环$Ø_K / p Ø_K$上的码$C$确定$K$上的格$Λ_\ell (C) $。如果$ p \nmid \ell$，则环$\R:=Ø_K / p Ø_K$同构于$\F_{p^2}$或$\F_p \times \F_p$。给定一个$C$ over $\R$的码，定义了相应格上的theta函数.这些theta系列$θ_{Λ_{\ell}(C)}$可以用$C$的完整权重枚举器来表示。我们证明，对于任何两个$\ell < \ell^\prime$，其对应的theta函数的第一个$\frac {\ell + 1} 4$项是相同的。此外，我们推测，对于$\ell > \frac {p(n+1)(n+2)} 2$有一个唯一的完整的重量枚举对应于一个给定的θ函数。我们验证了素数$p< 7$和$\ell \leq 59$的猜想。
英文摘要：
Let $\ell>0$ be a square-free integer congruent to 3 mod 4 and $Ø_K$ the ring of integers of the imaginary quadratic field $K=Q(\sqrt{-\ell})$. Codes $C$ over rings $Ø_K / p Ø_K$ determine lattices $Λ_\ell (C) $ over $K$. If $ p \nmid \ell$ then the ring $\R:=Ø_K / p Ø_K$ is isomorphic to $\F_{p^2}$ or $\F_p \times \F_p$. Given a code $C$ over $\R$, theta functions on the corresponding lattices are defined. These theta series $θ_{Λ_{\ell}(C)}$ can be written in terms of the complete weight enumerator of $C$. We show that for any two $\ell < \ell^\prime$ the first $\frac {\ell + 1} 4$ terms of their corresponding theta functions are the same. Moreover, we conjecture that for $\ell > \frac {p(n+1)(n+2)} 2$ there is a unique complete weight enumerator corresponding to a given theta function. We verify the conjecture for primes $p< 7$ and $\ell \leq 59$.
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